The authors establish several results for Banach spaces with the Bishop-Phelps-Bollobás point property for the numerical radius (BPBpp-nu).
We start by reviewing some conditions and terminology. Let \(X\) be a Banach space, the set of states of \(X\) is \[ \Pi(X) =\{(x,x^*)\in S_X\times S_{X^*} : x^*(x)=1\}. \] The numerical radius of \(T \in \mathcal{L}(X)\) and the numerical index of \(X\) are defined, respectively, by \[ v(T) = \sup\{|x^*(T(x))|:(x, x^*) \in \Pi(X)\} \] and \[ n(X) = \inf\{v(T):T\in L(X),\ \|T\| = 1\}. \]
Definition 2.1. We say that \(X\) has the Bishop-Phelps-Bollobás point property for the numerical radius (BPBpp-nu) if, given \(\epsilon > 0,\) there is \(\eta(\epsilon) > 0\) such that, whenever \(T \in \mathcal{L}(X)\) with \(v(T) = 1\) and \((x, x^*) \in \Pi(X)\) satisfy \(|x^*(T(x))| > 1-\eta(\epsilon)\), there exists \(S\in \mathcal{L}(X)\) with \(v(S) = 1\) such that \(|x^*(S(x))| = 1\) and \(\|S- T\| < \epsilon\).
The authors give a summary of known results (including a list of Banach spaces with the aforementioned property).
Let \(G\) be a Hausdorff topological group with the identity element \(e\) and \(T\) be a Hausdorff topological space. An action \((\cdot.\cdot)\) of \((G, T)\) is a continuous function from \(G\times T\) to \(T\) such that \((e,t)=t\) and \((g_1,(g_2,t))=(g_1g_2,t),\) for every \( g_1,g_2 \in G\) and \( t \in T\). The action is said to be transitive if \(T = \{(g, t): g\in G\}\) for every \(t\in T\), and said to be micro-transitive if \(\{(g, t): g\in U\}\) is a neighbourhood of \(t \in T \) for every \(t \in T,\) whenever \(U\) is a neighbourhood of \(e \in G\). The second numerical index of \(X\) is defined by \[ n'(X)=\max \{k \geq 0: k\|T+\mathcal{Z}(X)\| \leq v(T) \text{ for all } T\in L(X)\}, \] with \(\mathcal{Z}(X)\) denoting the set of all skew Hermitian operators on \(X\) and \(\|T+\mathcal{Z}(X)\| =\inf \{\|T+S\|:S \in \mathcal{Z}(X)\}\).
Given a Banach space \(X\), we may take the group of surjective isometries on \(X\) as the group \(G\) and \(S_X\) as the topological space \(T\). We then say that \(X\) (or the norm of \(X\)) is micro-transitive (respectively, transitive) if the canonical action is micro-transitive (respectively, transitive).
The main theorem of Section 2 is as follows.:
Theorem 2.2. If the norm of \(X\) is micro-transitive and \(n'(X) > 0\), then \(X\) satisfies the BPBpp-nu.
The proof of this theorem is given in Section 2 and it follows as a corollary that Hilbert spaces (both real or complex) also have BPBpp-nu.
An interesting result with a short and quite interesting proof formulates that, for Banach spaces with numerical index 1, the BPBpp-nu is equivalent to being one-dimensional.
Motivated by the restrictive behaviour of the Bishop-Phelps-Bollobás point property for the numerical radius, the authors investigate a weaker property, i.e., the \(\eta\) that appears in the definition of the BPBpp-nu depends on both a given \(\epsilon >0\) and also on the state \((x, x^*) \in \Pi(X).\) This property is denoted by \(L_{pp}\)-nu. They also review the strong subdifferentiability of a norm on \(X\) (SSD, for short) at \(x\in X\), meaning that \[ \lim_{t\to 0+} \frac{1}{t} (\| x+th\|-\|x\|)| \] exists uniformly for \(h \in B_X\).